Dirac delta as Kernel

 Let

$$\boxed{\delta(x,s):=\delta_s (x)=\delta_0 (x-s)}$$

Then we ask what is the analogue of Sturm-Liouville equation $Ly=f$ when

$$\boxed{G(x,s)=\delta(x,s)}$$

So, if we define 

$$\boxed{L(x)=-x\frac{d}{dx}}$$

we have 

$$L\delta=\delta.$$

Note that

$$\frac{x^n}{n!}\delta^{(n)}(x)=(-1)^n \delta(x),$$

$$\sum_{n=0}^\infty\frac{x^n}{n!}\delta^{(n)}(x)= \delta(x)\sum_{n=0}^\infty(-1)^n,$$

Borel/Abel summation of Grandi's series

$$\delta(x)=2\sum_{n=0}^\infty\frac{x^n}{n!}\delta^{(n)}(x)$$